Unit 1 of 7

Accumulation and foundations

  • 4 chapters
  • Chapters 1 to 4

These chapters build the definite integral from area. A rectangle, a limit of Riemann sums and the fundamental theorem of calculus give exact values, and linearity splits a sum into signed parts.

The first chapter reads an integral as a rate times a change in input, so a constant height gives a rectangle and the units of the answer follow. The second covers the area under a straight line with thin strips and takes the limit as their width goes to zero. The third lets the endpoint move, shows that the accumulated area has the integrand as its derivative and finds the area under a parabola by reversing the power rule. The fourth splits a polynomial into terms, keeps the sign of each contribution and checks the total with one antiderivative.

Every chapter will be part of one long video. The video is coming soon, and each chapter page already has the definitions and an example for every key term.

Area, Riemann sums, the fundamental theorem and linearity

  1. What an integral measures, starting with a rectangle

    What a definite integral measures, read through a constant height over an interval, with the integrand, the limits of integration, signed area, the accumulation of a rate and the units of an integral.

    Chapter 1

  2. The Riemann-sum limit for a changing height

    How Riemann sums over a regular partition estimate the area under a changing height, how left-endpoint and right-endpoint sums bracket it, and how their limit as the mesh width shrinks gives the integral.

    Chapter 2

  3. The fundamental theorem and the power rule for a parabola

    How the fundamental theorem of calculus links an accumulation function to its integrand, how the reverse power rule gives an antiderivative of a power, and why an indefinite integral carries a constant of integration.

    Chapter 3

  4. Linearity and signed contributions for a sum of terms

    How linearity of the integral, built from the sum rule and the constant multiple rule, splits a polynomial integral into signed contributions, and how the average value of a function turns the total into a height.

    Chapter 4